In a topological space, you want to guarantee that every convergent sequence has at most one limit. Which separation axiom is sufficient (and necessary) for this?
AT₀ (Kolmogorov) — there exists an open set distinguishing any two points
BT₁ (Fréchet) — for any two points, each has an open set not containing the other
CT₂ (Hausdorff) — any two distinct points have disjoint open neighborhoods
DT₄ (Normal) — any two disjoint closed sets can be separated by open sets
T₂ (Hausdorff) is exactly what guarantees unique limits. If xₙ → x and xₙ → y with x ≠ y, the disjoint open sets U ∋ x and V ∋ y (guaranteed by T₂) eventually exclude all terms of the sequence simultaneously — a contradiction. In T₁ spaces, the open sets separating two points may overlap, allowing a sequence to satisfy both neighborhoods. Disjointness is the critical word.
Question 2 Multiple Choice
In a T₁ space, every singleton set {x} is closed. Which argument correctly establishes this?
AT₁ implies T₂, and in Hausdorff spaces all singletons are closed by a separate theorem
BEvery singleton is open in T₁ spaces, so its complement is closed by definition
CFor every y ≠ x, the T₁ condition provides an open set containing y but not x; the complement of {x} is a union of such open sets, hence open — making {x} closed
DIn T₁ spaces the topology is discrete, so every set including {x} is both open and closed
The complement of {x} is X\{x} = ∪_{y≠x} U_y, where each U_y is an open set given by T₁ that contains y but not x. A union of open sets is open, so X\{x} is open, meaning {x} is closed. T₁ does not imply T₂, and T₁ spaces need not be discrete — in the cofinite topology on an infinite set, singletons are closed (complements are cofinite = open) but the space is T₁ and not T₂.
Question 3 True / False
Every metric space is a Hausdorff (T₂) topological space.
TTrue
FFalse
Answer: True
Given two distinct points x and y in a metric space with d(x,y) = r > 0, the open balls B(x, r/2) and B(y, r/2) are disjoint open neighborhoods. If some point z were in both, the triangle inequality gives r = d(x,y) ≤ d(x,z) + d(z,y) < r/2 + r/2 = r, a contradiction. This construction works in any metric space, so all metric spaces — including ℝⁿ, normed vector spaces, and function spaces with a metric — are Hausdorff.
Question 4 True / False
Nearly every Hausdorff (T₂) space is also a normal (T₄) space.
TTrue
FFalse
Answer: False
This is false. T₄ requires separating any two disjoint *closed sets* by disjoint open sets; T₂ only requires separating *points*. There exist Hausdorff spaces that are not normal — a classic example is the Sorgenfrey plane (ℝ² with the lower-limit topology on each coordinate). The two axes are disjoint closed sets that cannot be separated by disjoint open sets in that topology. The implications run T₄ → T₃ → T₂ → T₁ → T₀, but none of these arrows reverses.
Question 5 Short Answer
Explain why Urysohn's lemma makes the normal (T₄) separation axiom especially significant for analysis, beyond the purely topological separation of closed sets.
Think about your answer, then reveal below.
Model answer: Urysohn's lemma states that a space is T₄ if and only if any two disjoint closed sets A and B can be separated by a continuous real-valued function f with f(A)=0 and f(B)=1. This bridges the gap between purely topological separation (open sets) and the existence of continuous functions. Normality is the minimum condition needed to construct real-valued continuous functions with prescribed values on closed sets, which underlies results like the Tietze extension theorem (extending continuous functions from closed subsets) and partitions of unity used throughout differential geometry and functional analysis.
The key insight is that separation axioms are not just about keeping points or sets topologically distinct — the stronger axioms actually determine what continuous functions exist. T₂ ensures uniqueness of limits but says little about continuous functions from the space to ℝ. T₄ is the threshold at which the topological structure is rich enough to support continuous real-valued functions with precise controlled behavior on closed sets, making it the natural setting for many analytic constructions.